00000273.htm

256 APPENDIX TO CHAPTER IX

for any pair of distinct points (s, t) of I, one of which is not an extremity
of I. A subset of C is called a simple closed curve if it is the set of points
of a simple loop. Equivalent definitions are that a simple arc is a subset
homeomorphic to [0,1], and a simple closed curve a subset homeomorphic
to the unit circle U (9.5.7).

(Ap.4.1) The complement in C of a simple arc is connected (in other words,
a simple arc does not cut the plane).

Let y be a simple path defined in I, and let/be the continuous mapping
of y(I) onto I, inverse to y. Let a, b be two distinct points of C — y(I). By
(Ap.3.1), we have to prove that the restriction (p of z -> s_a> _b(z)/\s_0t _b(z)\ to y(I)
is inessential. But we can write (p = (<p ° y) °/; the continuous mapping cp ° y
of I into U is inessential (Ap.2.6), and so is therefore cp by (Ap.2.2).

(Ap.4.2) (The Jordan curve theorem) Let H be a simple closed curve in C.
Then:

(a) C — H has exactly two connected components, one of which is bounded
and the other unbounded.

(b) The frontier of every connected component 0/C — H is H.

(c) If y is a simple loop defined in I and such that y(l) = H, then j(x; y) = 0
if x is in the unbounded connected component 0/C — H, and j(x; y) = +1 if x
is in the bounded connected component 0/C — H.

The proof is done in several steps.

(Ap.4.2.1) We first prove (b) without any assumption on the number of
components of C — H. Let A be a connected component of C — H; as
C — H is open, we see as in (Ap.3.1) that the frontier of A is contained in H.
Let z e H, and let / be a homeomorphism of U onto H; let £ = e^ie e U
be such that/(Q = z. Let W be an arbitrary open neighborhood of z in C,
V c W a closed ball of center z; then there is a number CD such that
0 < co < n and that f(e^il) e V for 9 - co < t < 9 + co; let J be the image
of that interval by t-+f(e^lt); then the complement L of J in H is the
image by t ->f(e^u) of the compact interval [9 + co - 27t, 0 - co] (9.5.7),
and is a simple arc by (9.5.7). It follows from (Ap.4.1) that the open set
C - L ^ C - H is connected. Therefore (9.7.2) for any x e A c C - L,
there is a path y in C - L, defined in I = [a, b], such that y(a) = x, y(b) = z.



	256 APPENDIX TO CHAPTER IX for any pair of distinct points (s, t) of I, one of which is not an extremity of I. A subset of C is called a simple closed curve if it is the set of points of a simple loop. Equivalent definitions are that a simple arc is a subset homeomorphic to [0,1], and a simple closed curve a subset homeomorphic to the unit circle U (9.5.7).

	(Ap.4.1) The complement in C of a simple arc is connected (in other words, a simple arc does not cut the plane). Let y be a simple path defined in I, and let/be the continuous mapping of y(I) onto I, inverse to y. Let a, b be two distinct points of C — y(I). By (Ap.3.1), we have to prove that the restriction (p of z -> s_a> _b(z)/\s_0t _b(z)\ to y(I) is inessential. But we can write (p = (<p ° y) °/; the continuous mapping cp ° y of I into U is inessential (Ap.2.6), and so is therefore cp by (Ap.2.2).

	(Ap.4.2) (The Jordan curve theorem) Let H be a simple closed curve in C. Then: (a) C — H has exactly two connected components, one of which is bounded and the other unbounded. (b) The frontier of every connected component 0/C — H is H. (c) If y is a simple loop defined in I and such that y(l) = H, then j(x; y) = 0 if x is in the unbounded connected component 0/C — H, and j(x; y) = +1 if x is in the bounded connected component 0/C — H. The proof is done in several steps.

	(Ap.4.2.1) We first prove (b) without any assumption on the number of components of C — H. Let A be a connected component of C — H; as C — H is open, we see as in (Ap.3.1) that the frontier of A is contained in H. Let z e H, and let / be a homeomorphism of U onto H; let £ = e^ie e U be such that/(Q = z. Let W be an arbitrary open neighborhood of z in C, V c W a closed ball of center z; then there is a number CD such that 0 < co < n and that f(e^il) e V for 9 - co < t < 9 + co; let J be the image of that interval by t-+f(e^lt); then the complement L of J in H is the image by t ->f(e^u) of the compact interval [9 + co - 27t, 0 - co] (9.5.7), and is a simple arc by (9.5.7). It follows from (Ap.4.1) that the open set C - L ^ C - H is connected. Therefore (9.7.2) for any x e A c C - L, there is a path y in C - L, defined in I = [a, b], such that y(a) = x, y(b) = z.